In this Letter we report results from a precise measurement of the double-polarization observable \(E\) 
and helicity-dependent cross sections \(\sigma _{1/2}\)  and \(\sigma _{3/2}\)  for \(\pi ^0\pi ^0\)  pairs off protons and
neutrons at the Mainz MAMI accelerator [1]}. In the formalism for pseudoscalar meson pairs given in
[2]} this observable would be \(P_z^{\odot }\) . The definition is identical to the one for the observable
\(E\)  in single meson production which we use as abbreviation. For a circularly polarized photon beam and a longitudinally
polarized target, two different relative spin orientations, parallel or antiparallel, corresponding to the cross
sections \(\sigma _{3/2}\)  \((\uparrow \uparrow )\)  and \(\sigma _{1/2}\)  \((\uparrow \downarrow )\)  are possible which are
termed helicity-3/2 and helicity-1/2. This two configurations correspond for the excitation of nucleon resonances
to the electromagnetic couplings \(A_{3/2}\)  and \(A_{1/2}\)  listed in the Review of Particle Physics (PDG) [3]}.
They are related to the asymmetry \(E\)  by
\(E=\frac{\sigma _{1/2}-\sigma _{3/2}}{\sigma _{1/2}+\sigma _{3/2}}=\frac{\sigma _{1/2}-\sigma _{3/2}}{2\sigma _{0}}\;,\) 
